Locally Realized Covering Relation

Let L be a lattice (or a bounded lattice or a complemented lattice, etc.), and let C_L be the covering relation of L:

 C_L={(x,y) in L^2|x covers y or y covers x}.

Then C_L is locally realized provided that for every finite subset F of L, there is a finitely generated sublattice K of L, that contains F, for which C_K=C_L intersection K^2. It can be shown that C_L is locally realized if and only if there is a hyperfinitely generated sublattice L_nu of ^*L such that C_(L_nu)=C_(^*L) intersection L_nu^2. Using this characterization of locally realized covering relations, the following standard result can be proved using nonstandard methods:

Let L be a locally finite lattice in which the covering relation is locally realized, and let rho be the sublattice of L^2 which is generated by Delta_L union C_L. Then rho is a connected tolerance of L, and it is in fact the smallest locally subconnected (and locally connected) tolerance of L.

This entry contributed by Matt Insall (author's link)

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Burris, S. and Sankappanavar, H. P. A Course in Universal Algebra. New York: Springer-Verlag, 1981., M.; Kaiser, K.; and Insall, M. "Some Nonstandard Methods Applied to Distributive Lattices." Zeitschrifte für Mathematische Logik und Grundlagen der Mathematik 36, 123-131, 1990.Grätzer, G. Lattice Theory: First Concepts and Distributive Lattices. San Francisco, CA: W. H. Freeman, 1971.Grätzer, G. Universal Algebra, 2nd ed. New York: Springer-Verlag, 1979.Insall, M. "Some Finiteness Conditions in Lattices Using Nonstandard Proof Methods." J. Austral. Math. Soc. 53, 266-280, 1992.

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Locally Realized Covering Relation

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Insall, Matt. "Locally Realized Covering Relation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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